The discounted penalty function and the distribution of the total dividend payments in a multi-threshold Markovian risk model

In this thesis, we study the expected discounted penalty function and the total dividend payments in a risk model with a multi-threshold dividend strategy, where the claim arrivals are modeled by a Markovian arrival process (MAP) and the claim amounts are correlated with the inter-claim times. Systems of integro-differential equations in matrix forms are derived for the expected discounted penalty function and the moments of the discounted dividend payments prior to ruin. A recursive approach based on the integro-differential equations is then provided to obtain the analytical solutions. In addition to the differential approach, by employing some new obtained results in the actuarial literature, another recursive approach with respect to the number of layers is also developed for the expected discounted dividend payments. Examples with exponentially distributed claim amounts are illustrated numerically.